Concavity & the 2nd Derivative Test

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AP Calculus Section 3.4 Concavity and the Second Derivative Test

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Concavity & the 2nd Derivative Test Section 4

AP Exam Practice

Definition of Concavity Let f be differentiable on an open interval, I. The graph of f is concave up on I if f’’ is positive and concave down on I if f’’ is negative.

Test for Concavity Let f be a function whose 2nd derivative exists on an open interval, I. If f’’(x) > 0, then f is concave up. If f’’(x) < 0, then f is concave down.

Ex 1: Concavity Determine the open intervals on which the graph of is concave up or down.

Ex 2: Concavity Determine the open intervals on which is concave up or down.

HOMEWORK Pg 189 #11-21 odds

Point of Inflection Point at which concavity changes. Theorem: If (c, f(c)) is a point of inflection of the graph of f, then either f’’(c)=0 or f’’ does not exist at c.

Ex 1: Points of Inflection Determine the points of inflection and intervals of concavity for Determine the points of inflection and concavity for

Theorem: The 2nd Derivative Test Let f be a function such that f’’(c)=0 and the 2nd derivative of f exists on an open interval containing c. 1. If f’’(c) > 0, then f(c) is a relative minimum. 2. If f’’(c) < 0, then f(c) is a relative maximum. 3. If f’’(c) = 0, then the test fails.

Ex 2: Relative Extrema Find the relative extrema for

HOMEWORK Pg 189 #11-21 odds (find pts. of inflection), 27 – 40 odds